1013

Beautiful Arithmetic

Copyright c© 2001 Kevin Carmody

The Number Symphonies of Royal Heath: Symphony no. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2The Number Symphonies of Royal Heath: Symphony no. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3The Number Symphonies of Royal Heath: Symphony no. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The Number Symphonies of Royal Heath: Symphony no. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The Number Symphonies of Royal Heath: Symphony no. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5The Number Symphonies of Royal Heath: Symphony no. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5The Number Symphonies of Royal Heath: Symphony no. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Patterns using all the digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Kordemsky’s infinite rectangular array no. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Kordemsky’s infinite rectangular array no. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Kordemsky’s infinite triangular array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Digit properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Power properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Operations with five twos and four fours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

The Number Symphonies of Royal Heath: Symphony no. 1

1 × 1 = 111 × 11 = 121

111 × 111 = 123211111 × 1111 = 1234321

11111 × 11111 = 123454321111111 × 111111 = 12345654321

1111111 × 1111111 = 123456765432111111111 × 11111111 = 123456787654321

111111111 × 111111111 = 12345678987654321

1 =1 × 1

1=

12

12

121 =22 × 22

1 + 2 + 1=

222

22

12321 =333 × 333

1 + 2 + 3 + 2 + 1=

3332

32

1234321 =4444 × 4444

1 + 2 + 3 + 4 + 3 + 2 + 1=

44442

42

123454321 =55555 × 55555

1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1=

555552

52

12345654321 =666666 × 666666

1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1=

6666662

62

1234567654321 =7777777 × 7777777

1 + 2 + 3 + 4 + 5 + 6 + 7 + 6 + 5 + 4 + 3 + 2 + 1=

77777772

72

123456787654321 =88888888 × 88888888

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1=

888888882

82

12345678987654321 =999999999 × 999999999

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1=

9999999992

92

2


1 + 1 + 1 = 3 3 × 37 = 1112 + 2 + 2 = 6 6 × 37 = 2223 + 3 + 3 = 9 9 × 37 = 3334 + 4 + 4 = 12 12 × 37 = 4445 + 5 + 5 = 15 15 × 37 = 5556 + 6 + 6 = 18 18 × 37 = 6667 + 7 + 7 = 21 21 × 37 = 7778 + 8 + 8 = 24 24 × 37 = 8889 + 9 + 9 = 27 27 × 37 = 999

1 × 9 + 2 = 1112 × 9 + 3 = 111

123 × 9 + 4 = 11111234 × 9 + 5 = 11111

12345 × 9 + 6 = 111111123456 × 9 + 7 = 1111111

1234567 × 9 + 8 = 1111111112345678 × 9 + 9 = 111111111

123456789 × 9 + 10 = 1111111111

1 × 8 + 1 = 9 9 × 9 + 7 = 8812 × 8 + 2 = 98 9 × 98 + 6 = 888

123 × 8 + 3 = 987 9 × 987 + 5 = 88881234 × 8 + 4 = 9876 9 × 9876 + 4 = 88888

12345 × 8 + 5 = 98765 9 × 98765 + 3 = 888888123456 × 8 + 6 = 987654 9 × 987654 + 2 = 8888888

1234567 × 8 + 7 = 9876543 9 × 9876543 + 1 = 8888888812345678 × 8 + 8 = 98765432 9 × 98765432 0 = 888888888

123456789 × 8 + 9 = 987654321 9 × 987654321 − 1 = 8888888888

3


1 + 2 = 34 + 5 + 6 = 7 + 8

9 + 10 + 11 + 12 = 13 + 14 + 1516 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24

25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 3536 + 37 + 38 + 39 + 40 + 41 + 42 = 43 + 44 + 45 + 46 + 47 + 48

49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 57 + 58 + 59 + 60 + 61 + 62 + 6364 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 = 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80

81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 + 89 + 90 = 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99

1. In any line, the number of terms to the left of the equal sign is one more than the number of terms tothe right of the equal sign. The product of these two numbers is the middle term of the line.

2. The difference between any term and the one immediately below it is always equal between any twolines, and the difference increases by 2 from line to line.

3. The first term of every line is a perfect square.


32 + 42 = 52

102 + 112 + 122 = 132 + 142

212 + 222 + 232 + 242 = 252 + 262 + 272

362 + 372 + 382 + 392 + 402 = 412 + 422 + 432 + 442

552 + 562 + 572 + 582 + 592 + 602 = 612 + 622 + 632 + 642 + 652

782 + 792 + 802 + 812 + 822 + 832 + 842 = 852 + 862 + 872 + 882 + 892 + 902

1. In any line, the number of terms to the left of the equal sign is one more than the number of terms tothe right of the equal sign. The product of these two numbers is half the middle term of the line.

2. From the last term of a line to the first term of the next line, there is, instead of an increase by 1, anincrease by an odd integer which itself increases by 2 from line to line.

4


19 × 1 = 19 1 + 9 = 10 1 + 0 = 1 1089 × 1 = 1089 1 + 0 + 8 + 9 = 1819 × 2 = 38 3 + 8 = 11 1 + 1 = 2 1089 × 2 = 2178 2 + 1 + 7 + 8 = 1819 × 3 = 57 5 + 7 = 12 1 + 2 = 3 1089 × 3 = 3267 3 + 2 + 6 + 7 = 1819 × 4 = 76 7 + 6 = 13 1 + 3 = 4 1089 × 4 = 4356 4 + 3 + 5 + 6 = 1819 × 5 = 95 9 + 5 = 14 1 + 4 = 5 1089 × 5 = 5445 5 + 4 + 4 + 5 = 1819 × 6 = 114 11 + 4 = 15 1 + 5 = 6 1089 × 6 = 6534 6 + 5 + 3 + 4 = 1819 × 7 = 133 13 + 3 = 16 1 + 6 = 7 1089 × 7 = 7623 7 + 6 + 2 + 3 = 1819 × 8 = 152 15 + 2 = 17 1 + 7 = 8 1089 × 8 = 8712 8 + 7 + 1 + 2 = 1819 × 9 = 171 17 + 1 = 18 1 + 8 = 9 1089 × 9 = 9801 9 + 8 + 0 + 1 = 1819 × 10 = 190 19 + 0 = 19 1 + 9 = 10 1089 × 10 = 10890 1 + 0 + 8 + 9 + 0 = 18


17= .142857

142857 × 1 = 142857 142857 × 7 = 999999 999999 ÷ 9 = 111111142857 × 3 = 428571 428571 × 7 = 2999997 2999997 ÷ 9 = 333333142857 × 2 = 285714 285714 × 7 = 1999998 1999998 ÷ 9 = 222222142857 × 6 = 857142 857142 × 7 = 5999994 5999994 ÷ 9 = 666666142857 × 4 = 571428 571428 × 7 = 3999996 3999996 ÷ 9 = 444444142857 × 5 = 714285 714285 × 7 = 4999995 4999995 ÷ 9 = 555555


113

= .076923

76923 × 1 = 76923 76923 × 13 = 999999 999999 ÷ 9 = 11111176923 × 10 = 769230 769230 × 13 = 9999990 9999990 ÷ 9 = 111111076923 × 9 = 692307 692307 × 13 = 8999991 8999991 ÷ 9 = 99999976923 × 12 = 923076 923076 × 13 = 11999988 11999988 ÷ 9 = 133333276923 × 3 = 230769 230769 × 13 = 2999997 2999997 ÷ 9 = 33333376923 × 4 = 307692 307692 × 13 = 3999996 3999996 ÷ 9 = 444444

76923 × 2 = 153846 153846 × 13 = 1999998 1999998 ÷ 9 = 22222276923 × 7 = 538461 538461 × 13 = 6999993 6999993 ÷ 9 = 77777776923 × 5 = 384615 384651 × 13 = 4999995 4999995 ÷ 9 = 55555576923 × 11 = 846153 846153 × 13 = 10999989 10999989 ÷ 9 = 122222176923 × 6 = 461538 461538 × 13 = 5999994 5999994 ÷ 9 = 66666676923 × 8 = 615384 615384 × 13 = 7999992 7999992 ÷ 9 = 888888

5

Patterns using all the digits

123456789 × 1 = 123456789 987654321 × 1 = 987654321123456789 × 2 = 246913578 987654321 × 2 = 1975308642123456789 × 3 = 370370367 987654321 × 3 = 2962962963123456789 × 4 = 493827156 987654321 × 4 = 3950617284123456789 × 5 = 617283945 987654321 × 5 = 4938271605123456789 × 6 = 740740734 987654321 × 6 = 5925925926123456789 × 7 = 864197523 987654321 × 7 = 6913580247123456789 × 8 = 987654312 987654321 × 8 = 7901234568123456789 × 9 = 1111111101 987654321 × 9 = 8888888889123456789 × 10 = 1234567890 987654321 × 10 = 9876543210

12345679 × 1 = 12345679 (no 8) 12345679 × 9 = 11111111112345679 × 2 = 24691358 (no 7) 24691358 × 9 = 22222222212345679 × 3 = 37037037 37037037 × 9 = 33333333312345679 × 4 = 49382716 (no 5) 49382716 × 9 = 44444444412345679 × 5 = 61728395 (no 4) 61728395 × 9 = 55555555512345679 × 6 = 74074074 74074074 × 9 = 66666666612345679 × 7 = 86419753 (no 2) 86419753 × 9 = 77777777712345679 × 8 = 98765432 (no 1) 98765432 × 9 = 88888888812345679 × 9 = 111111111 111111111 × 9 = 999999999

157 × 28 = 4396198 × 27 = 5346297 × 18 = 5346138 × 42 = 5796483 × 12 = 57961738 × 4 = 6952186 × 39 = 72541963 × 4 = 7852

2 =134586729

3 =174695823

4 =157683942

5 =134852697

6 =176582943

7 =167582394

8 =254963187

9 =574296381

9 =9752410836

=9582310647

=9574210638

=7524908361

=5823906471

=5742906381

6

Kordemsky’s infinite rectangular array no. 1

1 3 5 7 9 11 13 15 17 19

1 4 7 10 13 16 19 22 25 28

1 5 9 13 17 21 25 29 33 37

1 6 11 16 21 26 31 36 41 46

1 7 13 19 25 31 37 43 49 55

1 8 15 22 29 36 43 50 57 64

1 9 17 25 33 41 49 57 65 73

1 10 19 28 37 46 55 64 73 82

1 11 21 31 41 51 61 71 81 91

1 12 23 34 45 56 67 78 89 100

For reference, we imagine each row being numbered 1, 2, 3, etc. from the top, and each column beingnumbered 1, 2, 3, etc. from the left. The right-angled corridors, called gnomons, we also imagine as beingnumbered 1, 2, 3, etc. from the upper left corner.

1. In a given row, the numbers are increasing by a fixed amount, which is one more than the row number.The same is true of the columns, except that the amound of increase is one less than the column number.

2. A number on the central diagonal is the square of its row or column number.

3. The sum of the numbers in a gnomon is the cube of its gnomon number.

4. If we have any square, one of whose diagonals is part of the central diagonal, then the sum of thenumbers in that square is a perfect square, and the number that it is the square of is the sum of the rowor column numbers which pass through the square.

7

Kordemsky’s infinite rectangular array no. 2

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 14 21 28 35 42 49 56 63 70

8 16 24 32 40 48 56 64 72 80

9 18 27 36 45 54 63 72 81 90

10 20 30 40 50 60 70 80 90 100

1. This is an ordinary multiplication table—that is, the number in a given spot is the product of its rownumber and its column number.

2. The number of each gnomon is the number on the outside of the gnomon, either at the left or on thetop.

3. The sum of the numbers in a gnomon is the cube of the gnomon number.

4. If we have any square which has one corner at the upper left corner of the array, then the sum of thenumbers in that square is a perfect square, and the number that it is the square of is the sum of the rowor column numbers which pass through the square.

8

5. The last two statements imply the following:

(1)2 = 13

(1 + 2)2 = 13 + 23

(1 + 2 + 3)2 = 13 + 23 + 33

(1 + 2 + 3 + 4)2 = 13 + 23 + 33 + 43

(1 + 2 + 3 + 4 + 5)2 = 13 + 23 + 33 + 43 + 53

(1 + 2 + 3 + 4 + 5 + 6)2 = 13 + 23 + 33 + 43 + 53 + 63

(1 + 2 + 3 + 4 + 5 + 6 + 7)2 = 13 + 23 + 33 + 43 + 53 + 63 + 73

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)2 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)2 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)2 = 13 + 23 + 33 + 43 + 53 + 63 + 73 + 83 + 93 + 103

Kordemsky’s infinite triangular array

1

2 3 4

5 6 7 8 9

10 11 12 13 14 15 16

17 18 19 20 21 22 23 24 25

26 27 28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

For reference, we imagine the rows being numbered 1, 2, 3, etc. from the top.

1. The square of each row number is the last number in the row.

2. If the number n is immediately over m, then n ×m is n rows directly below n.

3. If r is the row number, and c is the number in the exact center of the row, then c = r2 − r + 1, andc(r + 1) = r3 + 1.

9

Digit properties

37 + (3 × 7) = 32 + 72 111 × (11 + 1) = 113 + 13

37 × (3 + 7) = 33 + 73 147 × (14 + 7) = 143 + 73

48 × (4 + 8) = 43 + 83 148 × (14 + 8) = 143 + 83

37 × 1 = 037 37 × 2 = 074 37 × 4 = 148 37 × 5 = 185 37 × 7 = 259 37 × 8 = 29637 × 10 = 370 37 × 11 = 407 37 × 13 = 481 37 × 14 = 518 37 × 16 = 592 37 × 17 = 62937 × 19 = 703 37 × 20 = 740 37 × 22 = 814 37 × 23 = 851 37 × 25 = 925 37 × 26 = 962

9 + 9 = 18 24 + 3 = 27 47 + 2 = 49 497 + 2 = 4999 × 9 = 81 24 × 3 = 72 47 × 2 = 94 497 × 2 = 994

12 × 42 = 21 × 24 23 × 96 = 32 × 6912 × 63 = 21 × 36 24 × 63 = 42 × 3612 × 84 = 21 × 48 24 × 84 = 42 × 4813 × 62 = 31 × 26 26 × 93 = 62 × 3913 × 93 = 31 × 39 34 × 86 = 43 × 6814 × 82 = 41 × 28 36 × 84 = 63 × 4823 × 64 = 32 × 46 46 × 96 = 64 × 69

234256 = (2 + 3 + 4 + 2 + 5 + 6)4

23 + 42 + 56 = 112 = 65 + 24 + 32

09 = 32

1089 = 332

110889 = 3332

111088889 = 33332

11110888889 = 333332

16 = 42

1156 = 342

111556 = 3342

11115556 = 33342

1111155556 = 333342

49 = 72

4489 = 672

444889 = 6672

44448889 = 66672

4444488889 = 666672

81 = 92

9801 = 992

998001 = 9992

99980001 = 99992

9999800001 = 999992

10

Power properties

21 + 31 + 71 = 11 + 51 + 61

22 + 32 + 72 = 12 + 52 + 62

01 + 51 + 51 + 101 = 11 + 21 + 81 + 91

02 + 52 + 52 + 102 = 12 + 32 + 82 + 92

03 + 53 + 53 + 103 = 13 + 33 + 83 + 93

11 + 41 + 121 + 131 + 201 = 21 + 31 + 101 + 161 + 191

12 + 42 + 122 + 132 + 202 = 22 + 32 + 102 + 162 + 192

13 + 43 + 123 + 133 + 203 = 23 + 33 + 103 + 163 + 193

11 + 61 + 71 + 171 + 181 + 231 = 21 + 31 + 111 + 131 + 211 + 221

12 + 62 + 72 + 172 + 182 + 232 = 22 + 32 + 112 + 132 + 212 + 222

13 + 63 + 73 + 173 + 183 + 233 = 23 + 33 + 113 + 133 + 213 + 223

14 + 64 + 74 + 174 + 184 + 234 = 24 + 34 + 114 + 134 + 214 + 224

15 + 65 + 75 + 175 + 185 + 235 = 25 + 35 + 115 + 135 + 215 + 225

42 + 52 + 62 = 22 + 32 + 82

422 + 532 + 682 = 242 + 352 + 862

422 + 582 + 632 = 242 + 852 + 362

432 + 522 + 682 = 342 + 252 + 862

432 + 582 + 622 = 342 + 852 + 26z2

482 + 522 + 632 = 842 + 252 + 362

482 + 532 + 622 = 842 + 352 + 262

131 + 421 + 531 + 571 + 681 + 971 = 791 + 861 + 751 + 351 + 241 + 311

132 + 422 + 532 + 572 + 682 + 972 = 792 + 862 + 752 + 352 + 242 + 312

133 + 423 + 533 + 573 + 683 + 973 = 793 + 863 + 753 + 353 + 243 + 313

11

Operations with five twos and four fours

1 = 2 + 2 − 2 − 22

1 =44× 4

42 = 2 + 2 + 2 − 2 − 2 2 =

44+

44

3 = 2 + 2 − 2 +22

3 =4 + 4 + 4

44 = 2 × 2 × 2 − 2 − 2 4 = 4 + (4 − 4) × 4

5 = 2 + 2 + 2 − 22

5 =4 × 4 + 4

46 = 2 + 2 + 2 + 2 − 2 6 = 4 +

4 + 44

7 =222− 2 − 2 7 = 4 + 4 − 4

48 = 2 × 2 × 2 + 2 − 2 8 = 4 + 4 + 4 − 4

9 = 2 × 2 × 2 +22

9 = 4 + 4 +44

10 = 2 + 2 + 2 + 2 + 2 10 =44 − 4

411 =

222

+ 2 − 2

12 = 2 × 2 × 2 + 2 + 2

13 =22 + 2 + 2

214 = 2 × 2 × 2 × 2 − 2

15 =222

+ 2 + 2

16 = (2 × 2 + 2 + 2) × 2

17 = (2 × 2)2 +22

18 = 2 × 2 × 2 × 2 × +2

19 = 22 − 2 − 22

20 = 22 + 2 − 2 − 2

21 = 22 − 2 +22

22 = 22 × 2 − 22

23 = 22 + 2 − 22

24 = 22 + 2 + 2 − 2

25 = 22 + 2 +22

26 = 2 ×(

222

+ 2)

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1013